**5. Example**

Here we give a toy example that will allow the reader to understand the essence of the proposed approach. To begin the practical implementation of our approach, it is necessary to determine the specific values of the isotone valuation *v* and the metric *ρ (*see Eq. (7)–(9)). For brevity, we denote trapezoidal fuzzy numbers by *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *ai :* We will use the following isotone valuation *<sup>v</sup> <sup>R</sup>*<sup>~</sup> � � <sup>¼</sup> <sup>P</sup><sup>4</sup> *<sup>i</sup>*¼<sup>1</sup>*ai*. It can be easily shown that this valuation satisfies the conditions of Eq. (7) and (8). From this it follows that distance between two trapezoidal fuzzy numbers *<sup>R</sup>*~1, <sup>¼</sup> ð Þ *ai* and *<sup>R</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *bi* is determined as follows:

$$\rho\left(\tilde{R}\_1, \tilde{R}\_2\right) = \sum\_{i=1}^{4} |a\_i - b\_i|. \tag{25}$$

So, we have the following input: three experts' estimates of the parameter out of the components of the linguistic variable *A* "revenue," the result of aggregation *R*~ of the collection of three experts' estimates for this parameter, threshold criteria value *A*<sup>3</sup> selected from Eq. (18), and coordinates *k, t* specified by the manager (group of

Step 0: *Initialization: the result of aggregation of the collection of expert estimates for parameter "revenue"*– *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ <sup>1</sup>*:*037, 2*:*204, 3*:*028, 3*:*<sup>555</sup> *, threshold criteria value lender'<sup>s</sup> managers "not more than medium risk", coordinates*: 10, *t*<sup>2</sup> = 30, *t*<sup>4</sup> = 70, *k*<sup>2</sup> = 40, *k*<sup>3</sup> = 60*.* Step 1: *Compute* ∇, *Δ by Eq.* (26), *Eq.* (21) *and λ by Eq.* (22), ∇ = 1, *Δ* = 6, *λ =* 0.05. Step 2: *Compute trapezoidal fuzzy number, corresponding to the component A*<sup>3</sup> - the

**Summary.** In this section a toy example of the practical application of the proposed approach is provided. The concrete isotone valuation and metric are considered. We calculate the risk level for one parameter based on the estimates of three experts. For other parameters and any number of experts, the process will be similar.

The presented work aims to propose a new approach for an assessment of the credit risks under uncertainty. The novelty of the proposed approach is the use of trapezoidal fuzzy numbers, which makes it possible to adequately form and process the experts' estimates. An important fact is that the proposed approach takes into

• A brief analysis of existing models is carried out, and the feasibility of creating

• The rationale for the presentation of experts' assessments of the credit risk in

• A polar percentage and coordinate scales of trapezoidal fuzzy numbers with a gradation of assigned levels are defined. The formalization of the mapping of

managers) of the lender for use in Eq. (23).

*Graphical expression of the degree of risk in percent.*

*DOI: http://dx.doi.org/10.5772/intechopen.93285*

**Algorithm 4.**

**Figure 5.**

**6. Conclusions**

**193**

We follow Algorithm 2, detailing it along the way.

*A New Approach for Assessing Credit Risks under Uncertainty*

degree of *risk is medium by Eq*. (23) *<sup>A</sup>*~<sup>3</sup> <sup>¼</sup> ð Þ <sup>2</sup>*:*5, 3, 4, 4*:*<sup>5</sup> .

• the condition is satisfied and the level of risk is acceptable.

Step 3: *Verification of the condition R*~ ⪯ *A*~3:

account the degrees of experts' importance. The main results of the work are as follows:

the form of trapezoidal fuzzy numbers is given.

• The linguistic variable "degree of credit risk" is formed.

the percentage scale to the coordinate scale is given.

a new approach is justified.

Without loss of generality, we consider the process of determining the degree of risk for one parameter evaluated by three experts. For any other parameter, the procedure described below is similar. As noted above, the parameterization of the risk assessment process is carried out by the lender manager. Suppose that for evaluation the parameter "*p*<sup>1</sup> – revenue" has been selected.

We ask three experts to evaluate the parameter *p*<sup>1</sup> in the form of trapezoidal fuzzy number. As a result, we obtain

$$\{\tilde{R}\_j\} = \{ (1, 2, 3, 3.5), (1, 2.5, 2.8, 3), (1.5, 3, 4, 6) \}\tag{26}$$

We follow Algorithm 1, detailing it along the way. **Algorithm 3.**

Step 0: *Initialization: the regulation of finite collection of trapezoidal fuzzy numbers R*~0 *j* n o <sup>¼</sup> f g ð Þ 1, 2, 2*:*8, 3 , 1, 2 ð Þ *:*5, 3, 3*:*<sup>5</sup> , 1ð Þ *:*5, 3, 4, 6 . *Denote the aggregation weight of the j-th expert by ω<sup>j</sup> and the final result by R*~.

Step 1: *By Eq.* (14) *computes the representative <sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>¼</sup> ð Þ <sup>1</sup>*:*1053, 2*:*2105, 3*:*0526, 3*:*<sup>6315</sup> *:* Step 2: *Do* Step 3 *forj* ¼ 1, 3*.*

Step 3: *By Eq.* (15) *compute* <sup>Δ</sup> *<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>R</sup>*<sup>~</sup> <sup>∗</sup> , *R*~ *<sup>j</sup>* � �: <sup>Δ</sup>1 = 0.499, <sup>Δ</sup>2 = 1.2789, <sup>Δ</sup><sup>3</sup> = 4.5. *Since all* Δ*<sup>j</sup>* > 0 *then by Eq.* (15) *ω*<sup>1</sup> = 0.666, *ω*<sup>2</sup> = 0.26, *ω*<sup>3</sup> = 0.074 *and by Eq.* (16) *we obtain the finite result*:

$$
\tilde{R} = (1.037, 2.204, 3.028, 3.555) \tag{27}
$$

Now we form the percentage scale of the linguistic variable *A* (see Eq. (18)). At stage 1 of the previous section, a graduation of this scale is given in general form (see **Figure 4**). Suppose lender managers have determined the lower and upper bases of trapezoidal fuzzy numbers corresponding to the components of the linguistic variable *A*, i.e., definition and confidence areas as a percentage:

*A*1—the degree of risk is negligible [0, 30], [0, 20]. *A*2—the degree of risk is low [10, 50], [20, 40]. *A*3—the degree of risk is medium [30, 70], [40, 60]. *A*4—the degree of risk is high [50, 90], [60, 80]. *A*5—the degree of risk is extreme [70, 100], [80, 100].

So, *t*<sup>1</sup> = 10, *t*<sup>2</sup> = 30, *t*<sup>3</sup> = 50, *t*<sup>4</sup> = 70, *t*<sup>5</sup> = 90, *k*<sup>1</sup> = 20, *k*<sup>2</sup> = 40, *k*<sup>3</sup> = 60, and *k*<sup>4</sup> = 80. Thus, **Figure 4** will be converted to the form shown in **Figure 5**:

*A New Approach for Assessing Credit Risks under Uncertainty DOI: http://dx.doi.org/10.5772/intechopen.93285*

#### **Figure 5.**

**5. Example**

*Banking and Finance*

Here we give a toy example that will allow the reader to understand the essence of the proposed approach. To begin the practical implementation of our approach, it is necessary to determine the specific values of the isotone valuation *v* and the metric *ρ (*see Eq. (7)–(9)). For brevity, we denote trapezoidal fuzzy numbers by

*i*¼1

Without loss of generality, we consider the process of determining the degree of risk for one parameter evaluated by three experts. For any other parameter, the procedure described below is similar. As noted above, the parameterization of the risk assessment process is carried out by the lender manager. Suppose that for

We ask three experts to evaluate the parameter *p*<sup>1</sup> in the form of trapezoidal

Step 0: *Initialization: the regulation of finite collection of trapezoidal fuzzy numbers*

Step 1: *By Eq.* (14) *computes the representative <sup>R</sup>*<sup>~</sup> <sup>∗</sup> <sup>¼</sup> ð Þ <sup>1</sup>*:*1053, 2*:*2105, 3*:*0526, 3*:*<sup>6315</sup> *:*

, *R*~ *<sup>j</sup>* � �

Now we form the percentage scale of the linguistic variable *A* (see Eq. (18)). At stage 1 of the previous section, a graduation of this scale is given in general form (see **Figure 4**). Suppose lender managers have determined the lower and upper bases of trapezoidal fuzzy numbers corresponding to the components of the linguistic variable *A*, i.e., definition and confidence areas as a percentage:

So, *t*<sup>1</sup> = 10, *t*<sup>2</sup> = 30, *t*<sup>3</sup> = 50, *t*<sup>4</sup> = 70, *t*<sup>5</sup> = 90, *k*<sup>1</sup> = 20, *k*<sup>2</sup> = 40, *k*<sup>3</sup> = 60, and *k*<sup>4</sup> = 80.

*Since all* Δ*<sup>j</sup>* > 0 *then by Eq.* (15) *ω*<sup>1</sup> = 0.666, *ω*<sup>2</sup> = 0.26, *ω*<sup>3</sup> = 0.074 *and by Eq.* (16) *we*

¼ f g ð Þ 1, 2, 2*:*8, 3 , 1, 2 ð Þ *:*5, 3, 3*:*5 , 1ð Þ *:*5, 3, 4, 6 . *Denote the aggregation weight of the*

� � <sup>¼</sup> f g ð Þ 1, 2, 3, 3*:*<sup>5</sup> , 1, 2 ð Þ *:*5, 2*:*8, 3 , 1ð Þ *:*5, 3, 4, 6 (26)

shown that this valuation satisfies the conditions of Eq. (7) and (8). From this it follows that distance between two trapezoidal fuzzy numbers *<sup>R</sup>*~1, <sup>¼</sup> ð Þ *ai* and

� � <sup>¼</sup> <sup>P</sup><sup>4</sup>

*ai* � *bi* j j*:* (25)

: Δ1 = 0.499, Δ2 = 1.2789, Δ<sup>3</sup> = 4.5.

*<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ <sup>1</sup>*:*037, 2*:*204, 3*:*028, 3*:*<sup>555</sup> (27)

*<sup>i</sup>*¼<sup>1</sup>*ai*. It can be easily

*<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *ai :* We will use the following isotone valuation *<sup>v</sup> <sup>R</sup>*<sup>~</sup>

*ρ R*~1, *R*~<sup>2</sup>

evaluation the parameter "*p*<sup>1</sup> – revenue" has been selected.

We follow Algorithm 1, detailing it along the way.

*A*1—the degree of risk is negligible [0, 30], [0, 20]. *A*2—the degree of risk is low [10, 50], [20, 40]. *A*3—the degree of risk is medium [30, 70], [40, 60]. *A*4—the degree of risk is high [50, 90], [60, 80]. *A*5—the degree of risk is extreme [70, 100], [80, 100].

Thus, **Figure 4** will be converted to the form shown in **Figure 5**:

� � <sup>¼</sup> <sup>X</sup><sup>4</sup>

*<sup>R</sup>*~<sup>2</sup> <sup>¼</sup> ð Þ *bi* is determined as follows:

fuzzy number. As a result, we obtain

**Algorithm 3.**

*obtain the finite result*:

*R*~0 *j* n o

**192**

*R*~ *j*

*j-th expert by ω<sup>j</sup> and the final result by R*~.

Step 3: *By Eq.* (15) *compute* <sup>Δ</sup> *<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup> <sup>R</sup>*<sup>~</sup> <sup>∗</sup>

Step 2: *Do* Step 3 *forj* ¼ 1, 3*.*

*Graphical expression of the degree of risk in percent.*

So, we have the following input: three experts' estimates of the parameter out of the components of the linguistic variable *A* "revenue," the result of aggregation *R*~ of the collection of three experts' estimates for this parameter, threshold criteria value *A*<sup>3</sup> selected from Eq. (18), and coordinates *k, t* specified by the manager (group of managers) of the lender for use in Eq. (23).

We follow Algorithm 2, detailing it along the way. **Algorithm 4.**

Step 0: *Initialization: the result of aggregation of the collection of expert estimates for parameter "revenue"*– *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> ð Þ <sup>1</sup>*:*037, 2*:*204, 3*:*028, 3*:*<sup>555</sup> *, threshold criteria value lender'<sup>s</sup> managers "not more than medium risk", coordinates*: 10, *t*<sup>2</sup> = 30, *t*<sup>4</sup> = 70, *k*<sup>2</sup> = 40, *k*<sup>3</sup> = 60*.*

Step 1: *Compute* ∇, *Δ by Eq.* (26), *Eq.* (21) *and λ by Eq.* (22), ∇ = 1, *Δ* = 6, *λ =* 0.05. Step 2: *Compute trapezoidal fuzzy number, corresponding to the component A*<sup>3</sup> - the degree of *risk is medium by Eq*. (23) *<sup>A</sup>*~<sup>3</sup> <sup>¼</sup> ð Þ <sup>2</sup>*:*5, 3, 4, 4*:*<sup>5</sup> .

Step 3: *Verification of the condition R*~ ⪯ *A*~3:

• the condition is satisfied and the level of risk is acceptable.

**Summary.** In this section a toy example of the practical application of the proposed approach is provided. The concrete isotone valuation and metric are considered. We calculate the risk level for one parameter based on the estimates of three experts. For other parameters and any number of experts, the process will be similar.

#### **6. Conclusions**

The presented work aims to propose a new approach for an assessment of the credit risks under uncertainty. The novelty of the proposed approach is the use of trapezoidal fuzzy numbers, which makes it possible to adequately form and process the experts' estimates. An important fact is that the proposed approach takes into account the degrees of experts' importance.

The main results of the work are as follows:


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